Prof. Eric Steinhart (C) 1999
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I cite sections of Leibniz's Monadology in quotes, then follow them with comments. The translation is by Robert Latta (1898) with my emendations; Latta's translation is, to the best of my knowledge, in the public domain. This is a work in progress.
36. "But there must also be a sufficient reason for contingent truths or truths of fact, that is to say, for the sequence or connexion of the things which are dispersed throughout the universe of created beings, in which the analyzing into particular reasons might go on into endless detail, because of the immense variety of things in nature and the infinite division of bodies. There is an infinity of present and past forms and motions which go to make up the efficient cause of my present writing; and there is an infinity of minute tendencies and dispositions of my soul, which go to make its final cause."
36. Monads are infinitely complex; each monad is an infinitesimal unity containing an infinity of information, and transforming all that information in an infinitesimal instant of time (change is continuous). I don't know how to make a computational model in which anything is infinitely complex, or in which there are any actual infinities at all, so we leave the complexity of monads finite. Our model explicitly fails to satisfy Leibniz's Monadology in this respect. But this failure illuminates an important issue: is infinite complexity even logically possible?
37. "And as all this detail again involves other prior or more detailed contingent things, each of which still needs a similar analysis to yield its reason, we are no further forward: and the sufficient or final reason must be outside of the sequence or series of particular contingent things, however infinite this series may be."
38. "Thus the final reason of things must be in a necessary substance, in which the variety of particular changes exists only eminently, as in its source; and this substance we call God."
39. "Now as this substance is a sufficient reason of all this variety of particulars, which are also connected together throughout; there is only one God, and this God is sufficient."
40. "We may also hold that this supreme substance, which is unique, universal and necessary, nothing outside of it being independent of it,- this substance, which is a pure sequence of possible being, must be illimitable and must contain as much reality as is possible."
41. "Whence it follows that God is absolutely perfect; for perfection is nothing but amount of positive reality, in the strict sense, leaving out of account the limits or bounds in things which are limited. And where there are no bounds, that is to say in God, perfection is absolutely infinite. (Theod. 22, Pref. [E. 469 a; G. vi. 27].)"
42. "It follows also that created beings derive their perfections from the influence of God, but that their imperfections come from their own nature, which is incapable of being without limits. For it is in this that they differ from God. An instance of this original imperfection of created beings may be seen in the natural inertia of bodies. (Theod. 20, 27-30, 153, 167, 377 sqq.)"
43. "It is farther true that in God there is not only the source of existences but also that of essences, in so far as they are real, that is to say, the source of what is real in the possible. For the understanding of God is the region of eternal truths or of the ideas on which they depend, and without Him there would be nothing real in the possibilities of things, and not only would there be nothing in existence, but nothing would even be possible. (Theod. 20.)"
44. "For if there is a reality in essences or possibilities, or rather in eternal truths, this reality must needs be founded in something existing and actual, and consequently in the existence of the necessary Being, in whom essence involves existence, or in whom to be possible is to be actual. (Theod. 184-189, 335.)"
45. "Thus God alone (or the necessary Being) has this prerogative that He must necessarily exist, if He is possible. And as nothing can interfere with the possibility of that which involves no limits, no negation and consequently no contradiction, this [His possibility] is sufficient of itself to make known the existence of God a priori. We have thus proved it, through the reality of eternal truths. But a little while ago we proved it also a posteriori, since there exist contingent beings, which can have their final or sufficient reason only in the necessary Being, which has the reason of its existence in itself."
46. "We must not, however, imagine, as some do, that eternal truths, being dependent on God, are arbitrary and depend on His will, as Descartes, and afterwards M. Poiret, appear to have held. That is true only of contingent truths, of which the principle is fitness [convenance] or choice of the best, whereas necessary truths depend solely on His understanding and are its inner object. (Theod. 180-184, 185, 335, 351, 380.)"
47. "Thus God alone is the primary unity or original simple substance, of which all created or derivative Monads are products and have their birth, so to speak, through continual fulgurations of the Divinity from moment to moment, limited by the receptivity of the created being, of whose essence it is to have limits. (Theod. 382-391, 398, 395.)"
48. "In God there is Power, which is the source of all, also Knowledge, whose content is the variety of the ideas, and finally Will, which makes changes or products according to the principle of the best. (Theod. 7, 149, 150.) These characteristics correspond to what in the created Monads forms the ground or basis, to the faculty of Perception and to the faculty of Appetition. But in God these attributes are absolutely infinite or perfect; and in the created Monads or the Entelechies (or perfectihabiae, as Hermolaus Barbarus translated the word) there are only imitations of these attributes, according to the degree of perfection of the Monad. (Theod. 87.)"
49. "A created thing is said to act outwardly in so far as it has perfection, and to suffer [or be passive, patir] in relation to another, in so far as it is imperfect. Thus activity [action] is attributed to a Monad, in so far as it has distinct perceptions, and passivity [passion] in so far as its perceptions are confused. (Theod. 32, 66, 386.)"
50. "And one created thing is more perfect than another, in this, that there is found in the more perfect that which serves to explain a priori what takes place in the less perfect, and it is on this account that the former is said to act upon the latter."
51. "But in simple substances the influence of one Monad upon another is only ideal, and it can have its effect only through the mediation of God, in so far as in the ideas of God any Monad rightly claims that God, in regulating the others from the beginning of things, should have regard to it. For since one created Monad cannot have any physical influence upon the inner being of another, it is only by this means that the one can be dependent upon the other. (Theod. 9, 54, 65, 66, 201. Abrege, Object. 3.)"
49-51. Action among created beings is phenomenal (i.e. physical), not computational. Monads are all symmetrically coordinated, none acts on any other. However, their bodies, which are patterns on their VR details, act on one another (e.g. a glider acts on an eater when it crashes into the eater). As bodies move, their motions are changes in the mind/body relation of the monad. This relation is the mapping of the VR detail onto the sensory inputs (the affective fields) of the monad's AI subprogram (this is abbreviated as the VR-AI mapping). Phenomenally, the motion of the glider and the functionality of the eater as a phenomenal pattern are the reasons for what happens to the glider. First the glider acts on the eater (by crashing into it), and then the eater acts on the glider (by consuming it). But this isn't the action of one monad on another monad, except insofar as the glider is the body of one monad and the eater is the body of another monad. Reasons for the interactions of bodies are found in the changes in the VR detail of the monads involved. But these reasons are just the ones given in the physical explanations of changes in phenomena. Changes in the VR-AI mappings in one monad are explainable in terms of the changes in the VR-AI mappings in the monads with whose bodies it is interacting.
52. "Accordingly, among created things, activities and passivities are mutual. For God, comparing two simple substances, finds in each reasons which oblige Him to adapt the other to it, and consequently what is active in certain respects is passive from another point of view; active in so far as what we distinctly know in it serves to explain [rendre raison de] what takes place in another, and passive in so far as the explanation [raison] of what takes place in it is to be found in that which is distinctly known in another. (Theod. 66.)"
52. Each monad has the logical right to exist with all and only those monads that are spatially and temporally consistent with itself. So, every monad logically demands that if God actualizes it, then God ought to actualize all and only those that are spatially and temporally consistent with it. So each monad demands that God actualizes some spatio-temporally consistent set of monads that contains it. Now we need to define spatio-temporal consistency for sets of monads.
Monads have qualities. Qualities have values. Quality Qi of monad M is Qi(M). The value of a quality Qi of monad M is V(Qi(M)).
The state S(M) of monad M pairs each quality Qi of M with some value of that quality: S = { (Q1, 0), (Q2, 1), . . .}.
A transformation of a monad is a function from the qualities of the monad to the qualities of the monad. F = { (Q1, Q2), (Q2, Q3), . . . } is the successor transformation. I = { (Q1, Q1), (Q2, Q2), . . . } is the identity transformation.
A transformation is natural if and only if it preserves the geometric order of the VR detail in the monad; that is, if and only if it preserves the neighbor relation of qualities. Transformation F is natural if and only if for every pair of qualities Qi and Qj, F(Qi) is the neighbor of F(Qj) if and only if Qi is the neighbor of Qj. If qualities are cells in a grid, shifts, rotations, and reflections all preserve the structure of the grid. Unnatural transformations of qualities require that qualities be removed from their neighborhoods by tearing the grid. Figure 1 shows 4 natural transformations.
Figure 1. Four natural transformations of a grid.
Any series of natural transformations is a natural transformation. If F(x) is a natural transformation and G(x) is a natural transformation, then the composition G(F(x)) is a natural transformation. Figure 2 shows a series of three natural transformations.
Figure 2. A series of natural transformations is also natural.
State S(A) of monad A corresponds to state S(B) monad B if and only if there is some natural transformation F that pairs each quality Qi of A with exactly one corresponding quality F(Qi) of B such that Qi(A) has the same value as (F(Qi))(B).
Correspondence (of states) is an equivalence relation: it is reflexive, symmetrical, and transitive. If state S(A) of monad A corresponds to state S(B) monad B, then S(A) and S(B) are able to coexist in the same world. States that do not correspond are not able to coexist, because the monads with those states cannot be adjusted to one another.
A series of states and transitions {S1(A), S2(A), . . . Sn(A)} of monad A corresponds with a series of states and transitions {S1(B), S2(B), . . . Sn(B)} of monad B if and only if each state Si(A) corresponds to state Sj(B) if i = j. Any series of states and transitions of a monad is part of its history; so, parts of histories of monads are able to correspond.
Monads A and B are coordinated for a period of duration D if and only if there is some moment x in A and some moment y in B and some natural transformation F such that S(A,x) = F(S(B,y)) and for all t < D, S(A, x+t) = F(S(B, y+t)) and S(A, x-t) = F(S(B, y-t)). The period (x-t, x+t) of A is coordinated with period (y-t, y+t) of B.
Monads A and B are coordinated sempiternally if and only if there is some moment x in A and some moment y in B and some natural transformation F such that S(A,x) = F(S(B,y)) and for all t > 0, S(A, x+t) = F(S(B, y+t)) and S(A, x-t) = F(S(B, y-t)). If two monads A and B are coordinated sempiternally, then they are coordinated. Coordination is an equivalence relation (it is reflexive, symmetric, and transitive). All the monads in any possible world must be coordinated. Monads A and B are coordinated by F if they are coordinated and F is the natural transformation that coordinates them.
The moment x in A and moment y in B are their separate moments of coordination. If A and B are coordinated, then let x in A and y in B be the common moment 0. This moment is external to A and to B; it is the zero moment for the pair A and B. It is the moment at which A and B are synchronized; synchronized moments x+t and y+t for all t are simultaneous. Simultaneity is a relation between monads. If a set S of monads is coordinated, then there is a set of moments (one for each monad in S) at which they are synchronized; each moment in the set is the 0 moment for its monad and is the common moment for the set of monads; it is the 0 moment of the whole set.
A set of monads S is harmonious if and only if every pair of monads in S is coordinated. All pairs of monads in a harmonious set do not have to be coordinated by the same natural transformation: the set of monads {A, B, C} is harmonious if the pair (A, B) is coordinated by F, pair (A, C) is coordinated by G, and (B, C) is coordinated by H. Any possible world is a harmonious set of monads. Harmony ensures that there is some common moment 0 for all the monads in the set; it is what places all the monads in the same world-time.
Any two monads A and B might be coordinated by accident. But God does not make worlds accidentally; all the monads in a world must be in it for some reason. If any monads are in a world, there is some reason that suffices to explain why they are in it. One reason that is sufficient for putting two monads in a world is that they have the same nature: they are the same kind or type of monad.
Two monads are of the same type if and only if the nature of the one is the same as the nature of the other. The nature of any monad is its detail (its set of interrelated qualities). So two monads P and Q are of the same type if and only if there is some way to associate each quality p of P with exactly one quality q of Q such that if p has some relation R with other qualities of P then q has a corresponding relation R with corresponding qualities of Q. Since all relations are the same, two monads of the same type run the same program (but they are not the same because they run it on different data). Since the natures are the same, the monads can be coordinated. However, this does not imply that they are coordinated.
A set of monads is uniform if and only if all pairs of monads in the set are of the same type. Every possible world is a uniform set of monads. Every harmonious set of monads is uniform, but not every uniform set is harmonious. Uniformity is necessary but not sufficient for harmony. Uniformity says that monads are possible coordinated; harmony says that they are actually coordinated.
Unformity is necessary but not sufficient for possible worlds. Worlds must also be complete. Completeness is defined in terms of closure under a set of transformations. If the application of an operation to an object in some set yields another object in that set, the set is said to be closed under that operation. For example, the integers are closed under addition, since addition of two integers always yields another integer; but the integers are not closed under division, since 4 divided by 5 is not an integer. The only operations on monad are natural transformations, so closure for sets of monads is defined in terms of natural transformations.
A set S of monads is closed under a set of natural transformations N if and only if for every monad A in S and for every F in N there is some other monad B in S such that A is coordinated with B by F. If N is the set of all possible natural transformations for the monads of some type, then any set that is closed under N is naturally closed.
A set S of monads is symphonic if and only if S is uniform (so that all monads in S are of the type T) and S is harmonic and naturally closed under the natural transformations of T. Every possible world is some symphonic set of monads. Symphony is necessary and sufficient for being a possible world. Symphony is worldliness.
Every symphonic set of monads is a spatio-temporally consistent set of monads. So each monad logically demands that if God actualizes it, then God actualizes it in the context of some symphonic world. But symphony is merely formal: it does not determine the content of any world, so it does not suffice to define a world any more than knowing that thing must be red tells you the particular identity of that thing. God must choose symphonic sets of monads for worlds, but we do not yet know how God is able to choose symphonic sets, or which sets are symphonic, since we don't know what God chooses those sets from.
53. "Now, as in the Ideas of God there is an infinite number of possible universes, and as only one of them can be actual, there must be a sufficient reason for the choice of God, which leads Him to decide upon one rather than another. (Theod. 8, 10, 44, 173, 196 sqq., 225, 414-416.)"
54. "And this reason can be found only in the fitness [convenance], or in the degrees of perfection, that these worlds possess, since each possible thing has the right to aspire to existence in proportion to the amount of perfection it contains in germ. (Theod. 74, 167, 350, 201, 130, 352, 345 sqq., 354.)"
55. "Thus the actual existence of the best that wisdom makes known to God is due to this, that His goodness makes Him choose it, and His power makes Him produce it. (Theod. 8, 78, 80, 84, 119, 204, 206, 208. Abrege, Object. 1 and 8.)"
53,54,55. Every possible world is a set of monads. Not every set of monads is a possible world. Since each monad demands that it exist with all and only those monads with which it is actively or passively coordinated, every possible world must be a symphonic.
Prior to selecting the best of all possible worlds, God must determine the set of possible worlds (since the best of all possible worlds is in that set). To determine the set of possible worlds: God determines the set of all possible types of monads; for each possible type of monad, God determines the set of possible worlds of that type; God takes the union of all possible worlds of all types.
The set of possible monads of any type is determined by the detail of that type. Let X**Y be X raised to the Y-th power; so: 3**2 is 3 squared, which is 9. If the detail has N qualities each of which is able to take on K values, there are K**N possible assignments of values to those qualities, hence K**N possible monads of that type.
For instance, suppose the detail is an n-by-n grid, so that N is the square of n. If the detail is a 4 by 4 grid; then n=4 and N=16. If any quality in the n-by-n grid is able to take on one of 2 values, then for an n-by-n 2-valued grid there are 2**N possible monads, each running the same program on a grid with N cells. So for any 4-by-4 2-valued grid, there are 2**16 possible monads.
A set of monads of some type is complete if and only if it contains all possible monads of that type. Thus a complete set of monads whose detail is a 4-by-4 grid contains 216 different monads. A complete set of monads with infinitely many qualities is itself infinite.
If the complete set of all monads of some type is M, then the set of all sets of monads of that type is the set of all subsets of M. The set of all subsets of a set S is the power set of S, denoted POW(S). For instance, the power set of {1,2,3} is {{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}. If M is a complete set of monads of some type (if M is a complete uniform set), then its power set is likewise POW(M).
Each member of the power set POW(M) is a set made of monads of the type in M. Each m in POW(M) is uniform; so m is symphonic if it is harmonious and naturally closed. For every type T of monads, let SYM(T) be the set of all symphonic sets of monads of that type. The set SYM(T) is the set of all possible worlds for monads of type T.
It is far from clear how to rank the perfection of possible worlds; one way is to rank them according to their physical richness. Possible worlds whose histories have more regularities (more patterns with more complex yet orderly behaviors) are worlds with richer physics. Worlds with richer physics have more order and variety. Physical richness is natural perfection. If U is more physically rich than V, then U is naturally and aesthetically more perfect than V.
Figure 3 shows four different fundamental physical laws able to be realized by a cellular automaton like the game of life. The numbers B and S stand for "Born" (a quality changes from 0 to 1) and "Survives" (a quality stays 1). So the game of life is B3/S23, meaning that a quality changes from 0 to 1 if and only if it has 3 neighbors that are 1, and stays 1 if and only if it has 2 or 3 neighbors that are 1. For each law, a mobile pattern is shown and the richness of the world is indicated in the comments. Here the game of life is the richest world.
Figure 3. Some principles of change and their perfections.
We can rank or evaluate worlds according to their degrees of perfection. From least perfect to most perfect, we can rank worlds as follows: worlds with no regularities; worlds with oscillating patterns; worlds with moving patterns; worlds with more complex machines; worlds with self-reproducing patterns; worlds with patterns that are universal computing machines. Worlds able to contain patterns that are universal computing machines are the physically richest and naturally most perfect of all; the physical Church-Turing thesis says that such worlds contain machines of arbitrarily high complexity.
Leibniz says there is some one best of all possible worlds. I have no idea why there is only one, or how to determine which one it is given his criteria for goodness. I say all possible worlds are actual.